Suppose the heights in inches of the students in your class are as follows: 58, 58, 59, 60, 62, 64, 64, 65, 66, 66, 66, 66, 68, 68, 69, 70, 71, 72, 72, 74, 75, 77. What would be the mean of this data? How about the median and mode? Would you be able to calculate the variance for this data? How about the standard deviation? After completing this Concept, you'll be able to calculate measures of central tendency and dispersion like these.

Guidance

The majority of this textbook centers upon
two-variable data
, data with an input and an output. This is also known as
bivariate data
. There are many types of situations in which only one set of data is given. This data is known as
univariate data
. Unlike data you have seen before, no rule can be written relating univariate data. Instead, other methods are used to analyze the data. Three such methods are the
measures of central tendency
.

Measures of central tendency
are the center values of a data set.

Mean is the average of all the data. Its symbol is
.

Mode is the data value appearing most often in the data set.

Median is the middle value of the data set, arranged in ascending order.

Example A

Mrs. Kramer collected the scores from her students test and obtained the following data:

The variance is a measure of the dispersion and its value is lower for tightly grouped data than for widely spread data. In the example above, the variance is 27. What does it mean to say that tightly grouped data will have a low variance? You can probably already imagine that the size of the variance also depends on the size of the data itself. Below we see ways that mathematicians have tried to standardize the variance.

The Standard Deviation

Standard deviation
measures how closely the data clusters around the mean. It is the square root of the variance. Its symbol is
.

Example C

Calculate the standard deviation of the previous data set.

Solution:

The standard deviation is the square root of the variance.

Video Review

Guided Practice

Find the mean, median, mode, range, variance, and standard deviation of the data set below.

If each score on an algebra test is increased by seven points, how would this affect the:

Mean?

Median?

Mode?

Range?

Standard deviation?

If each score of a golfer was multiplied by two, how would this affect the:

Mean?

Median?

Mode?

Range?

Henry has the following World History scores: 88, 76, 97, 84. What would Henry need to score on his fifth test to have an average of 86?

Explain why it is not possible for Henry to have an average of 93 after his fifth score.

The mean of nine numbers is 105. What is the sum of the numbers?

A bowler has the following scores: 163, 187, 194, 188, 205, 196. Find the bowler’s average.

Golf scores for a nine-hole course for five different players were: 38, 45, 58, 38, 36.

Find the mean golf score.

Find the standard deviation to the nearest hundredth.

Does the mean represent the most accurate center of tendency? Explain.

Ten house sales in Encinitas, California are shown in the table below. Find the mean, median, and standard deviation for the sale prices. Explain, using the data, why the
median house price
is most often used as a measure of the house prices in an area.

Address

Sale Price

Date Of Sale

643 3RD ST

$1,137,000

6/5/2007

911 CORNISH DR

$879,000

6/5/2007

911 ARDEN DR

$950,000

6/13/2007

715 S VULCAN AVE

$875,000

4/30/2007

510 4TH ST

$1,499,000

4/26/2007

415 ARDEN DR

$875,000

5/11/2007

226 5TH ST

$4,000,000

5/3/2007

710 3RD ST

$975,000

3/13/2007

68 LA VETA AVE

$796,793

2/8/2007

207 WEST D ST

$2,100,000

3/15/2007

Determine which statistical measure (mean, median, or mode) would be most appropriate for the following.

The life expectancy of store-bought goldfish.

The age in years of the audience for a kids' TV program.

The weight of potato sacks that a store labels as “5-pound bag.”

James and John both own fields in which they plant cabbages. James plants cabbages by hand, while John uses a machine to carefully control the distance between the cabbages. The diameters of each grower’s cabbages are measured, and the results are shown in the table. John claims his method of machine planting is better. James insists it is better to plant by hand. Use the data to provide a reason to justify
both sides
of the argument.

James

John

Mean Diameter (inches)

7.10

6.85

Standard Deviation (inches)

2.75

0.60

Two bus companies run services between Los Angeles and San Francisco. The mean journey times and standard deviations in those times are given below. If Samantha needs to travel between the cities, which company should she choose if:

She needs to catch a plane in San Francisco.

She travels weekly to visit friends who live in San Francisco and wishes to minimize the time she spends on a bus over the entire year.

Inter-Cal Express

Fast-dog Travel

Mean Time (hours)

9.5

8.75

Standard Deviation (hours)

0.25

2.5

Mixed Review

A square garden has dimensions of 20 yards by 20 yards. How much shorter is it to cut across the diagonal than to walk around two joining sides?

Rewrite in standard form:
.

Solve for
:
.

A sail has a vertical length of 15 feet and a horizontal length of 8 feet. To the nearest foot, how long is the diagonal?